We also recover this inequality from (REF ). This inequality and the inequalities (REF ) are known already. They appear in various forms in the work of Birman and Solomyak, in particular for dealing with Schrödinger operators on bounded domains under Neuman's boundary condition (see, e.g., [1]}, [2]}, [3]}). In most of the instances where it has been used by Birman and Solomyak, the inequality (REF ) is relevant, since in those situations \(\ker H=1\)  and \(\Pi _0V\Pi _0\)  has a negative eigenvalue is \(V\le 0\)  and \(V\ne 0\) . However, in general we might have \(N^{-}(H_{V})- \dim \ker H <0\)  (e.g., if \(V(\ker H)\subset \operatorname{ran}H\) ), and so in such a case (REF ) is not sharp (see Example REF  below).

